Finding all values $p$ for which $int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dx$ converges
$begingroup$
I've been stuck for a while with this exercise. Find all positive real values $p$ for which the integral
$$int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dx$$
converges. So far I've came up with this:
$$ int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dxgeint_e^{+infty} frac{1}{(1+x^3)^frac{1}{p}}dx
\
text{Take the limit of dividing the second function by }frac{1}{x^frac{3}{p}}
\
lim_{xto{+infty}}frac{x^frac{3}{p}}{(1+x^3)^frac{1}{p}}=1
\
int_e^{+infty}frac{1}{x^frac{3}{p}}text{ Diverges } leftrightarrow pge3
$$
So, when $pge3$ my p-series diverges, which means my lower boundary for the main function diverges, which implies that the main function diverges. But I'm unable to prove any other implication. Any suggestions?
calculus integration convergence improper-integrals
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
asked Nov 30 '18 at 0:20
twkmztwkmz
545
$endgroup$
add a comment |
$begingroup$
I've been stuck for a while with this exercise. Find all positive real values $p$ for which the integral
$$int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dx$$
converges. So far I've came up with this:
$$ int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dxgeint_e^{+infty} frac{1}{(1+x^3)^frac{1}{p}}dx
\
text{Take the limit of dividing the second function by }frac{1}{x^frac{3}{p}}
\
lim_{xto{+infty}}frac{x^frac{3}{p}}{(1+x^3)^frac{1}{p}}=1
\
int_e^{+infty}frac{1}{x^frac{3}{p}}text{ Diverges } leftrightarrow pge3
$$
So, when $pge3$ my p-series diverges, which means my lower boundary for the main function diverges, which implies that the main function diverges. But I'm unable to prove any other implication. Any suggestions?
calculus integration convergence improper-integrals
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
asked Nov 30 '18 at 0:20
twkmztwkmz
545
$endgroup$
add a comment |
$begingroup$
I've been stuck for a while with this exercise. Find all positive real values $p$ for which the integral
$$int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dx$$
converges. So far I've came up with this:
$$ int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dxgeint_e^{+infty} frac{1}{(1+x^3)^frac{1}{p}}dx
\
text{Take the limit of dividing the second function by }frac{1}{x^frac{3}{p}}
\
lim_{xto{+infty}}frac{x^frac{3}{p}}{(1+x^3)^frac{1}{p}}=1
\
int_e^{+infty}frac{1}{x^frac{3}{p}}text{ Diverges } leftrightarrow pge3
$$
So, when $pge3$ my p-series diverges, which means my lower boundary for the main function diverges, which implies that the main function diverges. But I'm unable to prove any other implication. Any suggestions?
calculus integration convergence improper-integrals
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
asked Nov 30 '18 at 0:20
twkmztwkmz
545
$endgroup$
I've been stuck for a while with this exercise. Find all positive real values $p$ for which the integral
$$int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dx$$
converges. So far I've came up with this:
$$ int_e^{+infty} frac{ln(x)}{(1+x^3)^frac{1}{p}}dxgeint_e^{+infty} frac{1}{(1+x^3)^frac{1}{p}}dx
\
text{Take the limit of dividing the second function by }frac{1}{x^frac{3}{p}}
\
lim_{xto{+infty}}frac{x^frac{3}{p}}{(1+x^3)^frac{1}{p}}=1
\
int_e^{+infty}frac{1}{x^frac{3}{p}}text{ Diverges } leftrightarrow pge3
$$
So, when $pge3$ my p-series diverges, which means my lower boundary for the main function diverges, which implies that the main function diverges. But I'm unable to prove any other implication. Any suggestions?
calculus integration convergence improper-integrals
calculus integration convergence improper-integrals
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
asked Nov 30 '18 at 0:20
twkmztwkmz
545
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
asked Nov 30 '18 at 0:20
twkmztwkmz
545
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
edited Nov 30 '18 at 5:40
Olivier Oloa
108k17177294
108k17177294
asked Nov 30 '18 at 0:20
twkmztwkmz
545
asked Nov 30 '18 at 0:20
twkmztwkmz
545
asked Nov 30 '18 at 0:20
twkmztwkmz
545
545
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint. One may use that (with an integration by parts)
$$
int_e^infty frac{ln x}{x^a},dx qquad text{converges iff} qquad a>1.
$$ then one may observe that, as $x to infty$,
$$
frac{ln x}{(1+x^3)^frac{1}{p}} sim frac{ln x}{x^{3/p}}.
$$
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019439%2ffinding-all-values-p-for-which-int-e-infty-frac-lnx1x3-frac1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint. One may use that (with an integration by parts)
$$
int_e^infty frac{ln x}{x^a},dx qquad text{converges iff} qquad a>1.
$$ then one may observe that, as $x to infty$,
$$
frac{ln x}{(1+x^3)^frac{1}{p}} sim frac{ln x}{x^{3/p}}.
$$
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
$endgroup$
add a comment |
$begingroup$
Hint. One may use that (with an integration by parts)
$$
int_e^infty frac{ln x}{x^a},dx qquad text{converges iff} qquad a>1.
$$ then one may observe that, as $x to infty$,
$$
frac{ln x}{(1+x^3)^frac{1}{p}} sim frac{ln x}{x^{3/p}}.
$$
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
$endgroup$
add a comment |
$begingroup$
Hint. One may use that (with an integration by parts)
$$
int_e^infty frac{ln x}{x^a},dx qquad text{converges iff} qquad a>1.
$$ then one may observe that, as $x to infty$,
$$
frac{ln x}{(1+x^3)^frac{1}{p}} sim frac{ln x}{x^{3/p}}.
$$
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
$endgroup$
Hint. One may use that (with an integration by parts)
$$
int_e^infty frac{ln x}{x^a},dx qquad text{converges iff} qquad a>1.
$$ then one may observe that, as $x to infty$,
$$
frac{ln x}{(1+x^3)^frac{1}{p}} sim frac{ln x}{x^{3/p}}.
$$
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
answered Nov 30 '18 at 0:26
Olivier OloaOlivier Oloa
108k17177294
108k17177294
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019439%2ffinding-all-values-p-for-which-int-e-infty-frac-lnx1x3-frac1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
